Adaptive Fuzzy Robust Tracking Controller Design via Small Gain Approach and Its Application

Adaptive Fuzzy Robust Tracking Controller Design via Small Gain Approach and Its Application Yansheng Yang and Junsheng Ren Abstract—A novel adaptive fuzzy robust tracking control AFRTC

Adaptive Fuzzy Robust Tracking Controller Design via Small Gain Approach and Its Application

Yansheng Yang and Junsheng Ren

Abstract—A novel adaptive fuzzy robust tracking control

(AFRTC) algorithm is proposed for a class of nonlinear systems

with the uncertain system function and uncertain gain function,

which are all the unstructured (or nonrepeatable) state-dependent

unknown nonlinear functions arising from modeling errors

and external disturbances The Takagi–Sugeno type fuzzy logic

systems are used to approximate unknown uncertain functions

and the AFRTC algorithm is designed by use of the input-to-state

stability approach and small gain theorem The algorithm is

highlighted by three advantages: 1) the uniform ultimate

bound-edness of the closed-loop adaptive systems in the presence of

nonrepeatable uncertainties can be guaranteed; 2) the possible

controller singularity problem in some of the existing adaptive

control schemes met with feedback linearization techniques

can be removed; and 3) the adaptive mechanism with minimal

learning parameterizations can be obtained The performance

and limitations of the proposed method are discussed The uses

of the AFRTC for the tracking control design of a pole-balancing

robot system and a ship autopilot system to maintain the ship on a

predetermined heading are demonstrated through two numerical

examples Simulation results show the effectiveness of the control

scheme.

Index Terms—Adaptive robust tracking, fuzzy control,

input-to-state stability (ISS), nonlinear systems, small gain theorem.

I INTRODUCTION

IN RECENT years, interest in designing robust tracking

control for uncertain nonlinear systems has been ever

increasing, and many significant research attentions have been

attracted Most results addressing this problem are available

in the control literature, e.g., Kokotovic and Arcak [1] and

references therein And many powerful methodologies for

designing tracking controllers are proposed for uncertain

nonlinear systems The uncertain nonlinear systems may

be subjected to the following two types of uncertainties:

structured uncertainties (repeatable unknown nonlinearities),

which are linearly parameterized and referred to as parametric

uncertainties, and unstructured uncertainties (nonrepeatable

unknown nonlinearities), which are arising from modeling

errors and external disturbances To handle the parametric

uncertainties, adaptive control method, which has undergone

rapid developments in the past decade, e.g., [2]–[7] can be used

Manuscript received June 28, 2001; revised July 9, 2002 and January 15,

2003 This work was supported in part by the Research Fund for the Doctoral

Program of Higher Education under Grant 20020151005, the Science

Founda-tion under Grant 95-06-02-22, and the Young Investigator FoundaFounda-tion under

Grant 95-05-05-31 of the National Ministry of Communications of China.

The authors are with the Navigation College, Dalian Maritime University

(DMU), Dalian 116026, China (e-mail: ysyang@mail.dlptt.ln.cn).

Digital Object Identifier 10.1109/TFUZZ.2003.819837

As for unstructured uncertainties, if there is a prior knowledge

of the bounded functions, deterministic robust control method, e.g., [8]–[12] can be used Unfortunately, in industrial control environment, there are some controlled systems with the unstructured uncertainties where none of prior knowledge of the bounded functions is available, then the adaptive control method and the deterministic robust control method can not

be used to design controller for those systems A solution to that problem is presented that the neural networks (NNs) are used to approximate the continuous unstructured uncertain functions in the systems and Lyapunov’s stability theory is applied in designing adaptive NN controller Several stable adaptive NN control approaches are developed by [13]–[19] which guarantee uniform ultimate boundedness in the presence

of both unstructured uncertainties and unknown nonlinearities

As an alternative to NN control approaches, the intensive research has been carried out on fuzzy control for uncertain nonlinear systems The fuzzy systems are used to uniformly approximate the unstructured uncertain functions in the designed system by use of the universal approximation properties of the certain classes of fuzzy systems, which are proposed by [20] and [21], and a Lyapunov based learning law is used, and several stable adaptive fuzzy controllers that ensure the stability

of the overall system are developed by [22]–[26] Recently, an adaptive fuzzy-based controller combined with VSS and control technique has been studied in [27] and [28] However, there is a substantial restriction in the aforementioned works:

A lot of parameters are needed to be tuned in the learning laws when there are many state variables in the designed system and many rule bases have to be used in the fuzzy system for approximating the nonlinear uncertain functions,

so that the learning times tend to become unacceptably large for the systems of higher order and time-consuming process is unavoidable when the fuzzy logic controllers are implemented This problem has been pointed out in [26]

In this paper, we will present a novel approach for that problem A new systematic procedure is developed for the synthesis of stable adaptive fuzzy robust controller for a class

of continuous uncertain systems, and Takagi–Sugeno (T–S) type fuzzy logic systems [29] are used to approximate the un-known unstructured uncertain functions in the systems and the adaptive mechanism with minimal learning parameterizations can be achieved by use of input-to-state stability (ISS) theory first proposed by Sontag [31] and small gain approach given

in [32] The outstanding features of the algorithm proposed

in the paper are: i) that only one function is needed to be approximated by T–S fuzzy systems and no matter how many states in the designed system are investigated and how many

1063-6706/03$17.00 © 2003 IEEE

rules in the fuzzy system are used, only two parameters needed

to be adapted on-line, such that the burdensome computation of

the algorithm can be lightened increasingly and it is convenient

to realize this algorithm in engineering; and 2) the possible

controller singularity problem in some of the existing adaptive

control schemes met with feedback linearization techniques

can be avoided

This paper is organized as follows In Section II, we will

give the problem formulation, the description of a class of

nonlinear systems and tracking control problem of nonlinear

systems Section III contains some needed definitions of ISS,

small gain theorem and preliminary results In Section IV, a

systematic procedure for the synthesis of adaptive fuzzy robust

tracking controller (AFRTC) is developed In Section V, two

application examples for designing the tracking control for

the pole-balancing robot system and the ship autopilot system

by use of the AFRTC are included and numerical simulation

results are presented The final section contains conclusions

II PROBLEMFORMULATION

A System Description

Consider the th-order uncertain nonlinear systems of the

fol-lowing form:

(1)

where and represent the control input and the

output of the system, respectively

is comprised of the states which are assumed to be available, the

integer denotes the dimension of the system and

are unknown smooth uncertain functions and may contain

non-repeatable nonlinearities is the disturbance, unknown

but bounded, e.g., , where is an unknown

constant

Throughout this paper, the following assumption is made on

(1)

Assumption 1: The sign of is known, and there exists a

This assumption implies that smooth function is strictly

either positive or negative From now onwards, without loss of

As-sumption 1 is reasonable because being away from zero

is the controllable conditions of system (1) It should be

em-phasized that the low bound is only required for analytical

purposes, its true value is not necessarily known Some stability

is needed to proceed

Definition 1: It is said that the solution of (1) is uniformly

ultimately bounded (UUB) if for any , a compact subset of

We represent as any suitable vector norm In this paper,

vector norm is Euclidean, i.e., and given a

where denotes the operation of taking the max-imum (minmax-imum) eigenvalue The norm denoted by throughout this paper unless specified explicitly, is nothing but the vector two-norm over the space defined by stacking the ma-trix columns into a vector, so that it is compatible with the vector

The primary goal of this paper is to track a given reference signal while keeping the states and control bounded That

is, the output tracking error should be small The given reference signal is assumed to be bounded and has bounded derivatives up to the th order for all , and

is piecewise continuous

Sup-pose The (1) can be transformed into

(2)

In this paper, we present a method for the adaptive robust con-trol design for system (2) in the present of unstructured uncer-tainties Our design objective is to find an AFRTC of the form

(3) (4) where is the known fuzzy base functions In such a way that all the solutions of the closed-loop system (2)–(4) are uniformly ultimately bounded Furthermore, the output tracking error of the system can be steered to a small neighborhood of origin

B T–S Fuzzy Systems

In this section, we briefly describe the structure of fuzzy sys-tems Let denote the real numbers, the real -vectors, the real matrices Let be a compact simply connected set in With map , define to

be the function space such that is continuous A fuzzy system can be employed to approximate the function in order to design the adaptive fuzzy robust control law, thus the configu-ration of T–S type fuzzy logic system called T–S fuzzy system for short [29] and approximation theorem are discussed first as follows

Consider a T–S fuzzy system to uniformly approximate a continuous multidimensional function that has a com-plicated formulation, where is input vector with independent

The domain of is It fol-lows that the domain of is

In order to construct a fuzzy system, the interval [ ] is divided into subintervals

input fuzzy sets, denoted by , are defined

to fuzzify The membership function of is denoted by , which can be represented by triangular, trapezoid, gen-eralized bell or Gaussian type and so on

Generally, T–S fuzzy system can be constructed by the

constants The product fuzzy inference is employed to evaluate

theANDs in the fuzzy rules After being defuzzified by a typical

center average defuzzifier, the output of the fuzzy system is

(5)

, which is called a fuzzy base function When the membership function in

is denoted by some type of membership function, is

a known continuous function So, restructuring (5) as follows:

, . . . . , then

the (5) can be easily rewritten as

(6)

. . .

When the fuzzy system is used to approximate the continuous

function, two questions of interest may be asked: whether there

exists a fuzzy system to approximate any nonlinear function to

an arbitrary accuracy? how to determine the parameters in the

fuzzy system if such a fuzzy system does exist The following

lemma [30] gives a positive answer to the first question

Lemma 1: Suppose that the input universal of discourse is

a compact set in Then, for any given real continuous function

on and , there exists a fuzzy system in the

form of expression (6) such that

(7)

III MATHEMATICALPRELIMINARIES

The concept of ISS and ISS-Lyapunov function due to

Stontag [31], [33] and Sontag and Wang [34] have recently

been used in various control problems such as nonlinear

stabilization, robust control and observer designs (see, e.g.,

[35]–[40]) In order to ease the discussion of the design of

AFRTC scheme, the variants of those notions are reviewed

in the following First, we begin with the definitions of class , and functions which are standard in the stability literature; see [41]

Definition 2:

• A function is said of class if it is continuous, strictly increasing and It is of class

if it is of class and is unbounded

class if, for each , is of class , and, for each , is strictly decreasing and satisfies

, and is a class function if and only if there exist two class functions and such that

We consider the following system:

(8) where is the state and is the input For this system, we give the definition of input-to-state stable in the following

Definition 3: For (8), it is said to be input-to state practically

stable (ISpS) if there exist a function of class , called the nonlinear gain, and a function of class such that, for any initial condition , each measurable essentially bounded control defined for all and a nonnegative constant , the associated solutions are defined on [0, ) and satisfy

(9) where is the truncated function of at and stands for the supremum norm

When in (9), the ISpS property collapses to the ISS property introduced in [33]

Definition 4: A function is said to be an ISpS-Lya-punov function for (8) if

• there exist functions , of class such that

(10)

• there exist functions , of class and a constant

such that

(11) When (11) holds with , is referred to as an ISS-Lyapunov function

Then it holds that one may pick a nonlinear gain in (9)

of the form, which is given in [35]

(12) For the purpose of application studied in this paper, we intro-duce the sequel notion of exp-ISpS Lyapunov function

Definition 5: A function is said to be an exp-ISpS Lya-punov function for system (8) if

• there exist functions , of class such that

(13)

Fig 1 Feedback connection of composite systems.

• there exist two constants , and a class

function such that

(14)

When (14) holds with , the function is referred to as

an exp-ISS Lyapunov function

The three previous definitions are equivalent from [34] and

[39] Namely, the following

Proposition 1: For any control system (8), the following

properties are equivalent:

i) it is ISpS;

ii) it has an ISpS-Lyapunov function;

iii) it has an exp-ISpS Lyapunov function

Consider the stability of the closed-loop interconnection of

two systems shown in Fig 1

A trivial refinement of the proof of the generalized small

gain theorem given in [32] and [40] yields the following variant

which is suited for our applications here

Theorem 1: Consider a system in composite feedback form

(cf Fig 1)

(15) (16)

of two ISpS systems In particular, there exist two constants

, , and let and of class , and and

of class be such that, for each in the supremum

norm, each in the supremum norm, each and

de-fined on [0, ) and satisfy, for almost all

(17) (18) Under these conditions

(19) the solution of the composite systems (15) and (16) is ISpS

IV DESIGN OFADAPTIVEFUZZYROBUSTTRACKINGCONTROL

Using the pole-placement approach, we consider a term

where , the ’s are chosen such that all

left-half complex plane, leads to the exponentially stable dy-namics Then, the (2) can be transformed into

(20) where

. . . .

Because is stable, a positive–definite solution of the Lyapunov equation

(21) always exists and is specified by the designer

For this control problem, if both functions and in (20) are available for feedback, the technique of the feedback linearization can be used to design a well-defined controller, which is usually given in the form of

for some auxiliary control input with being nonzero for all time, such that the resulting closed-loop system can be shown to achieve a satisfactory tracking per-formance However, in many practical control systems, plant uncertainties that contain structured (or parametric) uncertain-ties and unstructured uncertainuncertain-ties (or nonrepeatable uncer-tainties) are inevitable Hence, both and may not

be available directly in the robust control design Obtaining a simple control algorithm as before is impossible Moreover,

if any adaptation scheme is implemented to estimate and as and respectively, the simple control algorithm aforementioned can be also used for substituting and for and , so the extra precaution

is required to guarantee that for all time At the present stage, no effective method is available in the litera-ture In this paper, we develop a semi-globally stable adaptive fuzzy robust controller which does not require to estimate the unknown function , and therefore avoids the possible controller singularity problem

In this paper, the effects due to plant uncertainties and external disturbances will be considered simultaneously The philosophy of our tracking controller design is expected that T–S fuzzy approximators equipped with adaptive algorithms are introduced first to learn the behaviors of uncertain dy-namics Here, only uncertain function is needed to be considered

For is an unknown continuous function, by Lemma 1, T–S fuzzy system with input vector for some compact set is proposed here to approximate the un-certain term where is a matrix containing the approxi-mating parameters Then, can be expressed as

(22) where is a parameter with respect to approximating accuracy

Substituting (22) into (20), we get

(23)

It follows that (23) reduces to

(24)

In order to design the adaptive fuzzy robust controller easily

by use of the small gain theorem, the following output equation

can be obtained by comparing (24) with (15):

Then, the feedback equation is given as follows:

So, (24) can be rewritten as (15) and (16)

(25) (26) Then, the feedback connection using the (25) and (26) can be

implemented using the block diagram shown in Fig 2

From Fig 2, we observe that the system should be made

to satisfy ISpS condition of the system through designing the

controller In (25), is an unknown, and there exist

some parameters with boundedness According to these

prop-erties, an adaptive fuzzy robust tracking control algorithm will

be proposed, which not only gives the controller to

make the system meet ISpS condition but also the online

adaptive law for and the other parameters in the (25) For this

purpose, we will discuss it in the following

Construct an adaptive fuzzy robust tracking controller as

follows:

(27) where denotes a certainty equivalent controller and

de-notes a supervisory controller for the disturbance,

approxima-tion error and other bounded items Those will be given in the

following

Substituting (27) into (25) yields

(28) Based on the aforementioned condition, we can get

(29)

the largest term with unknown constant in all boundedness In

order to design the controller, we can get

Fig 2 Feedback connection of fuzzy system.

Let and be the parameter estimate of and , respectively We propose an adaptive fuzzy robust tracking controller (AFRTC) as follows:

(30)

where will be specified by designer, and is the gain

of to be chosen later on

The adaptive laws for and are now chosen as

(31)

where , , 2 are the updating rates chosen by designer, and , , 2, and are design con-stants Adaptive laws (31) incorporate leakage term based on

a variant of the -modification proposed by Polycarpou and Ioannou [42], which can prevent parameter drift of the system

Theorem 2: Consider the system (20), suppose that

As-sumption 1 is satisfied and the can be approximated by

(21), then the control scheme (30) with adaptive laws (31) is

an AFRTC which can make all the solutions ( ) of the derived closed loop system uniformly ultimately bounded Furthermore, given any and bounds on and , we can tune our controller parameters such that the output error

Before proving Theorem 2, the following lemma given in [42]

is reviewed first

Lemma 2: The following inequality holds for any and

(32) where is a constant that satisfies , i.e.,

The proof of Theorem 2 can be divided into twofold First, let the constant and set as the input of the system , to prove the satisfaction of ISpS for the system by

use of the adaptive fuzzy robust tracking controller, and then to

prove uniform ultimate boundedness of the composite of two

systems with the feedback system by use of small

gain theorem

Choose the Lyapunov function as

(33)

The time derivative of along the error trajectory (28) is

(34)

We deal with relative items in (34), substitute (30) into the

relative items shown before, and obtain

(35)

(36) and substituting (30) into the relative items of (34), we get

Substituting (29) into the aforementioned equation yields

Let , by use of Lemma 2, the previous equa-tion can be rewritten as

(37) Substituting (35)–(37) into (34), such that

(38) Substituting (31) into (38), we get

(39)

, we get

By Definition 4, we propose the adaptive fuzzy robust tracking controller such that the requirement of ISpS for system can be satisfied with the functions and

of class By Definition 3 and the (12), we can get a gain function of system as follows:

For system , it is a static system such that we have

(40) Then, the gain function for system is According to the requirement of small gain Theorem 1, we can get

Owing to , the condition of the small gain

theorem 1 can be satisfied by choosing , so that it can be

proven that the composite closed-loop system is ISpS

There-fore, a direct application of Definition 3 yields that the

com-posite closed-loop system has bounded solutions over [0, )

More precisely, there exist a class -function and a

posi-tive constant such that

This, in turn, implies that the tracking error is bounded

over [0, ) By Proposition 1, there exists an ISpS-Lyapunov

function for the composite closed-loop system By substituting

(40) into (39), the ISpS-Lyapunov function is satisfied as

follows:

(41)

(41), we get

It results that the solutions of composite closed-loop system

are uniformly ultimately bounded, and implies that, for any

, there exists a constant such that for all The last statement of Theorem

2 follows readily since can be made arbitrarily

small if the design parameters , , , , are chosen

appropriately

Remark 1: It is interesting to note that most of the available

adaptive fuzzy controllers in the literature are based on feedback

linearization techniques, whose structures are usually taken the

of and , respectively, and be a new control variable

To avoid singularity problem when , several modified

adaptive methods were provided by [44], [25], and [28] In this

paper, the adaptive fuzzy robust tracking controller developed

before has the following properties:

where and According to those properties, it is easy

to show that we does not require to estimate the unknown gain

function In such a way we can not only reduce the number

of parameters needed to be adapted on-line for and but

also avoid the possible controller singularity problem usually

met with feedback linearization design when the adaptive fuzzy

control is executed

Remark 2: Since the function approximation property

of fuzzy systems is only guaranteed within a compact set, the stability result proposed in this paper is semiglobal in the sense that, for any compact set, there exists a controller with sufficiently large number of fuzzy rules such that all the closed-loop signals are bounded when the initial states are within this compact set In practical applications, the number of fuzzy rules usually can not be chosen too large due

to the possible computation problem This implies that the fuzzy system approximation capability is limited, that is, the approximating accuracy in (22) for the estimated function will be greater when chosen small number of fuzzy rules However, we can choose appropriately the design parameters , , , , to improve both stability and performance of the closed-loop systems

V APPLICATIONEXAMPLES

Now, we will reveal the control performance of the proposed AFRTC via application examples Two examples on designing tracking controller for pole-balancing robot system and ship au-topilot system are given in this section The former has an un-known input gain function and the latter unknown input gain constant We shall find the adaptive fuzzy robust tracking controllers by following the design procedures given in the pre-vious section Simulation results will be presented

A Pole-Balancing Robot System

To demonstrate the effectiveness of the proposed algorithms,

a pole-balancing robot is used for simulation The Fig 3 shows the plant composed of a pole and a cart The cart moves on the rail tracks in horizontal direction The control objective is to bal-ance the pole starting from an arbitrary condition by supplying a suitable force to the cart The same case studied has been given

in [43] The dynamic equations are described by

(42)

where

is the angular position from the equilibrium position and Suppose that the trajectory planning problem for a weight-lifting operation is considered and this pole-bal-ancing robot system suffers from uncertainties and exogenous disturbances The desired angle trajectory is assumed here

pendulum, is the mass of the vehicle, is the length of the pendulum and is the applied force Here, we use the

Fig 3 Pole-balancing robot system.

Define five fuzzy sets for each , with labels (NL),

(NM), (ZE), (PM), (PL) which are

character-ized by the following membership functions:

(43)

Twenty-five fuzzy rules for the fuzzy system are included in

the fuzzy rule bases Hence, the function is approximated

by T–S fuzzy system as follows:

(44)

where . , . . , can be defined

as (6)

of Lyapunov expression (21) is obtained by

If picking in (30), we can obtain the adaptive fuzzy

robust tracking controller for pole-balancing robot system as

follows:

(45)

(b) Fig 4 Simulation results for proposed AFRTC algorithm in this paper (a) Position of pole-balancing robot system (Solid line: actual position, Dashed line: reference position) (b) Control force.

Fig 5 Simulation results for the adaptive parameters when employing AFRTC algorithm (a) Adaptive parameter  (b) Adaptive parameter ^.

where

For the convenience of simulation, choose the initial

simula-tion results are shown in Figs 4 and 5

Before presenting the outstanding advantages of AFRTC

developed in this paper, we will briefly review the control

law proposed in [44] as follows:

(46)

Lya-punov equation and obtain

where if (which is a constant specified by the

Define five fuzzy sets the same as those in (43) for each ,

, twenty-five fuzzy rules for the fuzzy systems and

, respectively, and singleton fuzzifier, the product

infer-ence and the center-average defuzzification are used Hinfer-ence, the

functions and can be approximated by the fuzzy

with components

and

and the construction of is similar to

(b) Fig 6 Simulation results for Control algorithm in (46) (a) Position of pole-balancing robot system (Solid line: actual position, Dashed line: reference position) (b) Control force.

In Wang [44], use the following adaptive law to adjust pa-rameter vector ; see (47) at the bottom of the page, where the projection operation is defined as

In [44], use the following adaptive law to adjust parameter vector :

if

Here, for the parameters , , and , please refer to Wang [44] The simulation results are shown in Fig 6

Fig 7 shows the simulation results of tracking errors by use

of the proposed AFRTC and the controller given in (46), respec-tively From the results, we can see that the control performances are almost the same Hence, we can state that the AFRTC satis-fies the following advantages that have been described in Sec-tion IV: only one funcSec-tion is needed to be approximated

by T–S fuzzy systems and no matter how many states in the system are investigated and how many rules in the fuzzy system are used, only two parameters are needed to be adapted on-line

in AFRTC However, for the traditional methodology (e.g., the control law proposed in [44]), even based on five fuzzy sets for each state variable and singleton fuzzy model aforementioned, there are 50 parameters needed to be adapted online for the fuzzy system and when the fuzzy logic con-troller is implemented And the traditional methodology can cause the increase of the number of parameters needed to be

(b) Fig 7 Simulation results of the tracking errors (a) Proposed AFRTC

algorithm (b) Control algorithm in (46).

adapted exponentially with that of number of state variables

or fuzzy sets The computational complexity can be lessened

dramatically and the learning time can be reduced vastly when

using AFRTC developed in this paper Then AFRTC has the

potential to provide uncomplicated, reliable and

easy-to-under-stand solutions for a large variety of nonlinear control tasks even

for higher order systems

B Ship Autopilot System

Many of the present generation of autopilots installed in ships

are designed for the course keeping They aim at maintaining

the ship on a predetermined course and thus require directional

information Developments in the last 20 y include variants of

the analogue proportional-integral-derivative (PID) controller

In the recent years, some sophisticated autopilots are proposed

based on advanced control engineering concepts whereby the

gain settings for the proportional, derivative and integral terms

of heading are adjusted automatically to suit the dynamics of

the ship and environmental conditions such as model reference

adaptive control [46], self-tuning [47], optimal [48],

theo-ries [49] and adaptive robust fuzzy control [50]

In this paper, the adaptive fuzzy robust tracking controller

proposed above will be used for designing ship autopilot

Be-fore considering the designs of the autopilots, it is of interest to

describe the dynamics of the ship The mathematical model

re-lating the rudder angle to the heading of the ship is found to

be of the form

(49)

parame-ters which are function of ship’s constant forward velocity and

its length is a nonlinear function of The function

can be found from the relationship between and in steady

TABLE I

F UZZY IF – THEN R ULES

state such that An experiment known as the

“spiral test” has shown that can be approximated by

(50) where and are real valued constants

In normal steering, a ship often makes only small deviations from its desired direction The coefficient in the (50) could be equal to 0 such that a linear model is used as the design model for designing the autopilot, but in this paper, let both and be not equal to 0, a nonlinear model (50) is used as the design model for designing the adaptive fuzzy robust controller as following Let the state variables be , and control variable

be , then the (49) can be rewritten in the state–space form

(51)

Without loss of generality, we assume that the function

in the (50) can be defined in the function which is un-known with a continuous complicated formulation system func-tion, T–S fuzzy system can be constructed to approximate the function by the following nine fuzzyIF–THENrules

in Table I

denotes the fuzzy set “Positive”, denotes the fuzzy set “Zero” and denotes the fuzzy set “Negative” They can be character-ized by the membership functions as follows

For the previous example, we may use fuzzy sets on the normal-ized universes of discourse as shown in Fig 8

Using the center average defuzzifier and the product infer-ence engine, the fuzzy system is obtained as follows:

(52)

where . and . . can be defined

as the (6)

To demonstrate the availability of the proposed scheme, we take a general cargo ship with the length 126 m and the displace-ment 11 200 tons as an example for simulation